International Journal of Electronics and Communications. AEU-53 (6): pp. 1-17, December 1999
AES990806, MULTIFRAME DETECTOR/TRACKER: OPTIMAL PERFORMANCE, BRUNO & MOURA
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Multiframe Detector/Tracker: Optimal Performance
Marcelo G. S. Bruno(1) and Jos´e M. F. Moura(2) (1) Elect. Eng. Depart., University of S˜ ao Paulo, S˜ ao Paulo, SP 05508-090, Brazil ph: (55-11) 3818-5290; fax: (55-11) 3818-5128; email:
[email protected] (2) Depart. Elect. and Comp. Eng., Carnegie Mellon University, Pittsburgh, PA, 15213 ph: (412) 268-6341; fax: (412) 268-3890; email:
[email protected]
The first author was partially supported by CNPq-Brazil and FAPESP, S˜ ao Paulo, Brazil. This work was also funded by ONR grant N00014-97-1-0800
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Abstract We develop in this paper the optimal Bayes multiframe detector/tracker for rigid extended targets that move randomly in clutter. The performance of this optimal algorithm provides a bound on the performance of any other suboptimal detector/tracker. We determine by Monte Carlo simulations the optimal performance under a variety of scenarios including spatially correlated Gaussian clutter and non-Gaussian (K and Weibull) clutter. We show that, for similar tracking performance, the optimal Bayes tracker can achieve peak signal-to-noise ratio gains possibly larger than 10 dB over the commonly used combination of a spatial matched filter (spatial correlator) and a linearized Kalman-Bucy tracker. Simulations using real clutter data with a simulated target suggest similar performance gains when the clutter model parameters are unknown and estimated from the measurements. Keywords Multiframe Detection and Tracking; Nonlinear Stochastic Filtering; Noncausal Gauss-Markov Random Fields.
I. Introduction The paper studies integrated detection and tracking of randomly moving targets in clutter using as input data a sequence of noisy images. The images may be collected by electromagnetic sensors such as high resolution radars, or optical sensors such as infrared (IR) devices. At each sensor scan, an image or frame is produced. If one or more targets are present during a scan, the corresponding image contains the returns from the targets plus the returns from the background clutter. Otherwise, if no target is present, the sensor return consists exclusively of clutter. The clutter accounts for spurious reflectors, which may appear as false targets, and for measurement noise. In the case when multiple, at most M , targets of interest are present, the multitarget detector decides from the noisy data how many targets (0,1,2,...,M ) are present in each frame. Once a target is declared present by the detector, a subsequent tracker estimates its position in the surveillance space. The interpolation across successive scans of the estimated positions of a target forms a track for that target. Due to clutter, false detections, known as false alarms, may occur, and false tracks may be estimated. Conversely, actual targets may fail to be detected. This situation is known as a miss. Even if correct detections (i.e., no misses or false alarms) occur, the background clutter can still cause the tracker to produce a wrong estimate of the target’s position, i.e., a tracking error. The ultimate goal is to estimate the target state, typically a collection of kinematic components such as position, velocity, or acceleration. In most existing algorithms, e.g., [1], detection and tracking are two separate stages. The measurements of interest to the tracker are not the raw sensor images, but the outputs of preliminary detection subsystems. The detection stage involves the thresholding of the raw data, usually one single sensor frame. After further preprocessing, validated detections provide measurements that, for targets that are declared present, are treated as noise-corrupted observations of the target state such as, for example, direct estimates of position (range, azimuth, and elevation). Due to the ocurrence of random false alarms in the detection process, or due to clutter coming from spurious reflectors, interfering targets, or man-made decoys, validated measurements may actually be false measurements that do not originate from true targets. Multitarget trackers generally assume [1], [8] that the targets are pointwise and associate a linear (or linearized) dynamic model to the state of each target of interest. A tracking filter, usually a variation
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on the Kalman-Bucy filter, combines the validated measurements with the dynamic model, providing an estimate of the state of the target. An important issue arising from the decoupling of detection and tracking is the problem of deciding which set of measurements or weighted combination of measurements should be associated to each target state estimator or to clutter. This problem is known as data association. The most common data association algorithms, see [8], compute posterior probabilities of association conditioned on the measurements and use them throughout the estimation process. Brief review of the literature References concerned only with target detection, not tracking, include [4], [5], [6]. In [5], Pohlig introduces an algorithm for detection of constant velocity objects such as meteors, asteroids, and satellites, in fixed stellar backgrounds. The measurements are obtained by a staring sensor with an array of charged coupled device (CCD) sensors in the focal plane of a telescope. The focal plane image is integrated and sampled in space and time, resulting in a three-dimensional (two spatial dimensions and one temporal dimension) discrete model, where the optical intensity of both targets and the background are modeled as Poisson distributions with different means that reflect the different photon counts arising from targets and clutter. Pixel intensities under both hypotheses of presence and absence of target are assumed spatially uncorrelated. The detection algorithm in [5] is a 3D generalized likelihood ratio test (GLRT) based on batch processing: all available sensor frames are stacked in a data volume, and then the GLRT decides on the presence or absence of a target anywhere in that volume. The work by Reed, Gagliardi, and Shao [6] is similar in nature to Pohlig’s approach and introduces a 3D (again space plus time) matched filter for detection of known, moving targets within a Gaussian background clutter with known spectral density. However, unlike reference [5], reference [6] considers the case of continuous (non-sampled) data; it is best suited for optical rather than digital processing. A different problem is considered by Chen and Reed in [4]. The goal in [4] is to introduce a constant false alarm rate (CFAR) algorithm to solve the problem of detection of a known target signal in a given scene, using a set of K correlated reference scenes that contain no target or, alternatively, very weak target returns. The reference scenes are obtained either from different frequency bands of the main scene (multispectral or hyperspectral imagery) or from sequential observations in time. The proposed detection algorithm is a generalized likelihood ratio test (GLRT) that tests for the presence or absence of a target in the main scene using as data the entire collection of reference scenes plus the main scene itself. The underlying model assumes that, after pre-processing (essentially removal of the local variable mean), each individual scene is a zero-mean, Gaussian, white random vector, i.e., the spatial correlation between the pixels in each individual image is neglected. However, the model assumes a cross-correlation between pixels at the same spatial location in different scenes. An alternative modeling for multispectral imagery that incorporates both interframe and intraframe correlation was proposed in [7]. In this paper, instead of decoupling detection and tracking as in [1], or considering detection-only of moving objects as in [5], [6], we develop the optimal, multiframe, Bayes detector/tracker that processes directly the sensor images and integrates detection and tracking into a unified framework. The Bayesian strategy involves the computation at each scan of the posterior probability of the unknown target states conditioned on the observations. In [3], the author uses a dynamic programing approach and the Viterbi
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algorithm to study target detection. We postpone to section III-E a detailed discussion comparing the Bayes algorithm with the dynamic programming approach in [3]. In our approach, we integrate detection and tracking into the same framework by augmenting the target state space with additional dummy states that represent the absence of targets. The posterior probability of a given target being absent is propagated in time together with the posterior probabilities of the other “present target” states. In contrast to Pohlig’s batch detector [5], we develop a recursive framework where we still process all frames available in an optimal way, but these frames are processed one by one and discarded as we finish processing them. As a new frame is available, we simply update the posterior probabilities for the target states by running one more iteration of the algorithm. Modeling assumptions The optimal Bayesian algorithm takes full advantage of all prior information on the clutter, target signature, and target motion models, and allows multiframe detection and tracking with recursive processing across all observed sensor scans. We consider in this paper both pointwise (single pixel) and extended (multipixel) targets. We present detection results for targets with deterministic signatures and for targets with time-varying random signatures. The random signatures are described by multivariate, spatially correlated Gaussian distributions. We assume translational motions, and we define as the target state the spatial coordinates of the target’s geometric centroid. Since practical sensors have a finite resolution, we restrict the target centroid positions to a finite grid where each pixel represents a resolution cell of the sensor. We describe motions by finite state machines (FSMs) obtained by discretizing the continuous differential equations that describe the target dynamics. The dummy states that represent the absence of a target are incorporated into the FSM model that also specifies the transition probabilities between the absence and the presence of a target, and vice-versa. We consider two classes of clutter models: spatially correlated clutter with Gaussian statistics, and uncorrelated non-Gaussian clutter with heavy tail amplitude (envelope) statistics. The spatial correlation of the clutter is captured by using noncausal, spatially homogeneous, Gauss-Markov random fields (GMrfs) of arbitrary order [25]. GMrfs are statistical models that capture the locality properties of the clutter, namely, the clutter at a given spatial location is strongly dependent on the clutter intensity in neighboring locations. This assumption is intuitively realistic in many practical scenarios. Regarding non-Gaussian clutter, we represent it by spherically invariant-random vectors (SIRVs) [13], [14], [15], which have been shown to generate a variety of envelope statistics of practical interest, including the Weibull, K, Rician [18], and G [19] envelopes. Performance studies This paper focuses on performance results for the optimal multiframe Bayes detector/tracker in a variety of scenarios, including, as we mentioned before, both deterministic and random signature targets, observed in both Gaussian and non-Gaussian clutter. We test the proposed algorithm primarily on synthetic data with known clutter and target models. The optimal performance curves, obtained through extensive Monte Carlo simulations, provide an upper bound to the performance of suboptimal algorithms. We benchmark against these bounds the performance of competing suboptimal schemes such as the association of a single frame spatial correlator (matched filter) with a multiframe linearized Kalman-Bucy filter (KBf) tracker. These studies show that there is a significant margin of
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improvement to be had over existing detectors and trackers. In practice, the situation of perfect match between the data and the model is not realistic. In order to assess the robustness of the algorithm to mismatches between the measurements and the model, we present an example of detection/tracking with real clutter data, obtained by a laser radar mounted to the bottom of an aircraft. We fit the model to the real clutter by estimating its parameters from the data. The experimental results confirm that there is a significant improvement in performance over conventional algorithms such as a plain single frame image correlator associated with a KBf. Summary of the paper The paper is divided into 6 sections. Section I is this introduction. Section II presents the models for sensor, target, motion, and clutter that underly our integrated approach to detection and tracking. Section III examines the derivation of the optimal Bayesian detector/tracker based on the models from section II. Sections IV and V quantify respectively the detection and tracking performances of the algorithm through comprehensive Monte Carlo simulations assuming a single target scenario. Both correlated Gaussian clutter and non-Gaussian clutter situations are considered, and performance comparisons with alternative suboptimal detection and tracking algorithms are provided. Finally, section VI summarizes the contributions of the paper. We omit in this paper specific details on the implementation of the Bayes detector/tracker. These can be found in reference [24] for the particular case of a single, deterministic 2D target observed in GMrf clutter. II. The Model At each sensor scan, there are at most M targets present in the surveillance space. Each target is a rigid body with translational motion belonging to one of M possible classes characterized by their signature parameters, and by the dimensions of their noise-free image. We restrict our discussion to the situation where all targets are distinct. For simplicity of notation, we restrict this section to one-dimensional (1D) spaces. A brief discussion on the corresponding 2D models and a comprehensive investigation of 2D detection/tracking performance are found in section V (see also [24] for further details on modeling and implementation of the 2D detector/tracker algorithm). A. Surveillance Space and Target Model We first model the surveillance space of the sensor. Given the sensor’s finite resolution, we discretize the 1D space by the uniform finite discrete lattice L = {l: 1 ≤ l ≤ L} (1) where L is the number of resolution cells and l is an integer. We refer to the lattice L as the sensor lattice. The resolution cells are also referred to as pixels. To develop an integrated framework for detection and tracking, it is useful to extend the lattice L with additional states that will be used to represent the absence of targets and to account for the fact that target images extend over more than one pixel in the sensor lattice. We introduce first the vector T Zn = zn1 . . . znM , which collects the positions of the geometric centroids of the M possible targets in the sensor image at scan n. DRAFT
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Let the pixel length S p of a class p target, 1 ≤ p ≤ M , be S p = (lip + lsp + 1), where lip and lsp are the maximum extent in pixels of the target, respectively to the left and to the right of its centroid. These parameters are assumed known and time-invariant in the paper. If S p is odd, we make lip = lsp = (S p −1)/2. Otherwise, if S p is even, we adopt the convention that lsp = S p /2 and lip = lsp − 1. To account for the situations when targets move in and out of the sensor range, we define the extended centroid lattice, Lp = {l: − lsp + 1 ≤ l ≤ L + lip } (2) which corresponds to the set of all possible centroid positions znp such that at least one pixel of the target is still visible in the sensor image. Finally, to include the possibility of absence of a target, we introduce an additional dummy state. We adopt the convention that, whenever a class p target is not present at the nth scan, znp takes the value L + lip + 1. With the addition of this dummy absent target state, we define the augmented lattice, ep = {l: − lsp + 1 ≤ l ≤ L + lp + 1} . L (3) i p Extended Target Model When a class p target is present, i.e., zn ∈ Lp , the noise free target image is simply the spatial distribution of the target pixel intensities, apk , −lip ≤ k ≤ lsp , centered at the centroid lattice cell znp . Otherwise, if znp = L + lip + 1, meaning the target is absent, the sensor image corresponding to target-only returns reduces to a null image. These intuitive ideas are formalized mathematically by expressing the noise free image of a class p target at the nth sensor scan as the nonlinear function lp s X tp (znp ) = apk eznp +k znp ∈ Lp
(4)
k=−lp i
tp (znp ) = 0L znp = L + lip + 1 (5) where el , 1 ≤ l ≤ L, is an L-dimensional vector whose entries are all zero, except for the lth entry which is one. If l < 1 or l > L, el is defined as the identically zero vector. This particular definition for el outside the original sensor grid L is adopted to guarantee that the target model in (4) will accurately describe the disappearance of portions of the target from the sensor image as the target’s centroid moves closer to the boundaries of the surveillance space. The pixel intensity coefficients apk in (4) are also known as the target signature coefficients. They may be deterministic and known, deterministic and unknown, or random. Random signatures account for fluctuations in the reflectivity, or in the conditions of illumination of the target, as well as random variations in channel characteristics such as fading. For simplicity, we assume in most of this paper that the signature coefficients are known and time-invariant. An extension of the detection/tracking algorithms to targets with random signature in section III-D. Monte Carlo simulations with synthetic spatially correlated/temporally uncorrelated Gaussian targets are presented in section IV. B. Multitarget Observations and Clutter Models We consider a multitarget scenario with M possible targets, and collect the L sensor readings at each pixel of the nth scan in the L-dimensional column vector yn . Due to the presence of spurious reflectors and background, yn consists of the superposition of the various noise-free target images plus clutter. Using the extended target model introduced in subsection II-A, the observation vector or nth sensor frame, yn , is given by yn = t1(zn1 ) + t2(zn2 ) + . . . + tM (znM ) + vn
(6) DRAFT
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where vn is the background clutter vector, also referred to as the nth clutter frame, and tp(znp ), 1 ≤ p ≤ M , is given by equations (4) and (5), depending on whether the pth target is present or absent at the nth scan. The clutter vn is assumed to be statistically independent of tp (zn ), 1 ≤ p ≤ M . At each frame, the clutter at a given spatial location (pixel) may be statistically correlated with the clutter at another spatial location. The clutter intensity may also have Gaussian or non-Gaussian statistics. We adopt one of three models for vn : spatially white Gaussian clutter; spatially correlated Gaussian clutter; and spatially white non-Gaussian clutter. These models allow us to assess how clutter spatial correlation or non-Gaussian clutter statistics affect the performance of the detection/tracking algorithms. Gaussian clutter under the assumption of Gaussianity, the vector vn has a multivariate normal probability density function (pdf), p(vn) = N (0, R), where R is the clutter spatial covariance, and 0 is the mean. The zero mean assumption assumes a pre-processing stage that removes the possibly spatially variant local mean. A non-zero mean can be accounted for trivially. We distinguish two cases for the covariance matrix R. White spatially homogeneous Gauss clutter: With spatially uncorrelated (white) clutter, the covariance matrix R is diagonal. Assuming spatial homogeneity, R = σv2 I, where I is the identity matrix and σv2 is the variance (power) of the clutter. Spatially correlated homogeneous Gauss-Markov clutter: We model spatially correlated clutter as a Gauss-Markov random field (GMrf) [25]. This model simply states that the clutter intensity at a given pixel of the sensor image is a weighted average of the clutter intensity in neighboring pixels plus an error term. We assume in this paper a noncausal neighborhood region for each pixel. If we add the assumption of spatial homogeneity, an mth order 1D noncausal GMrf model for the nth clutter frame is given by the spatial difference equation vn (l) =
m X
αj [vn (l − j) + vn (l + j)] + un(l)
1≤l≤L
(7)
j=1
where un(l) is a zero-mean, correlated prediction error such that E [vn (l)un (k)] = 0
∀k 6= l
(8)
and the symbol E [.] stands for expectation or ensemble average. In order to completely define equation (7) at all pixel locations, we specify boundary conditions (bc’s) outside the sensor lattice L. Common boundary conditions are simply vn (l) = 0 for l < 1 or l > L. These are known as Dirichlet bc’s. Other bc’s can be alternatively used, see for example [23], [25]. Second-Order Statistics of GMrfs: The GMrf model is very attractive because it provides a simple parameterization for the inverse of the covariance matrix of the background clutter vn . Collecting the clutter samples vn (l) and the error samples un(l), 1 ≤ l ≤ L, in two L-dimensional vectors vn and un, an equivalent matrix representation for the difference equation in (7) is Avn = un
(9)
where A is a sparse and highly structured matrix usually referred to as the potential matrix. For the 1D mth order homogeneous model in (7), the potential matrix is an m-banded, Toeplitz, symmetric matrix
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with structure [25]
1
−α1
−α2
...
−αm
0
0
...
0
0
8
−α1 1 −α1 . . . −αm−1 −αm 0 ... 0 0 . .. .. .. .. .. .. .. .. .. .. A= (10) . . . . . . . . . . 0 0 0 . . . −αm−1 −αm−2 −αm−3 . . . 1 −α1 0 0 0 . . . −αm −αm−1 −αm−2 . . . −α1 1 We now derive the second-order statistics of un , which is referred to as the prediction error, and of the
clutter field vn . Combining the orthogonality condition in (8) with the matrix equation (9), we note that E unuTn = E Avn uTn = σu2 A (11) where the superscript “T” denotes the transpose of a vector or matrix. In (11), we used the assumption of spatial homogeneity (roughly speaking, the spatial “equivalent” of stationarity) to make E [vn (l)un (l)] = σu2 , for all l, 1 ≤ l ≤ L. Finally, since A is nonsingular and symmetric, then vn = A−1 un and −1 T −1 Σ−1 = A−1E un uTn A−T v = E vn vn −1 = σu2 A−1 AA−T = A/σu2 .
(12)
Equation (12) gives for free, with no matrix inversion required, the inverse of the clutter covariance in terms of the highly structured matrix A given in equation (10). This structure is used to design computationally efficient detection and tracking algorithms when the clutter is correlated as a Gauss-Markov random field. Finally, for our simulation studies, we use equation (9) and a technique based on the upper Cholesky factorization of the potential matrix A, [25], to generate samples of the GMrf clutter vn . Non-Gaussian clutter When dealing with non-Gaussian clutter, we assume that the sensor measures, at each resolution cell, the in-phase and quadrature returns of the clutter and targets echoes. The clutter measurements at instant n correspond to a sampling of the returned clutter complex envelope and are given by the even-sized vector
vn = vc1n vs1n . . . vcLn vsLn
(13)
where L is the number of resolution cells. We assume that the double-sized vector vn has a joint pdf with non-Gaussian statistics such that the sequence of random variables q ek = (vckn )2 + (vskn )2 1 ≤ k ≤ L
(14)
is identically distributed with a probability density function different from a Rayleigh distribution. K and Weibull envelope statistics We are interested in analyzing how the tracker performs against a background clutter whose envelope at each resolution cell has heavier tails than a Rayleigh envelope. Useful clutter envelope statistics are the K and Weibull models that are frequently used in the literature to represent the amplitude statistics of clutter returns [20], [21], [22]. The corresponding pdfs for the two models are [18] 1. K pdf: pE (e) =
bν+1 eν 2v−1 Γ(ν)
Kν−1 ( b e )
e≥0
where ν is a shape parameter, Γ(.) is the Eulerian function, Kν−1 ( . ) is a modified Bessel function of the second kind and b is related to σ2 by b2 = (2 ν)/σ2. 2. Weibull pdf: pE (e) = a c ec−1 exp(−a ec )
e≥0
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where c is a shape parameter and a relates to the average power σ of the quadrature components by 2σ2 = a−2/c Γ(1 + 2c ) . Simulation of K and Weibull clutter samples Rayleigh envelope statistics correspond to a multivariate joint Gaussian distribution of the in-phase and quadrature clutter returns. Similarly, heavy-tailed envelope statistics such as the Weibull and K distributions correspond to a generalized spherically-invariant random vector (SIRV) model in the backscatter domain [13], [14]. Techniques to simulate heavy-tailed clutter using SIRV models have been discussed extensively in the literature [14], [15], [16]. In particular, we used the algorithms in [16] to generate the samples of uncorrelated K and Weibull clutter that were used in the Monte Carlo simulations in section IV-B. We omit the simulation details here for lack of space and refer the reader instead to the literature, particularly [16]. C. Target Motion Assuming that the targets are rigid bodies with translational motion, the target motion is completely specified by the dynamics of the target centroid. We adopt a first order statistical model for the centroid dynamics. Given the sensor finite resolution, we model the motion of a class p target in the corresponding ep by a set of transition probabilities augmented lattice L p P (znp = k | zn−1 = j) k, j ∈ Lep . (15) p p The transition probabilities P (zn | zn−1), 1 ≤ p ≤ M , represent the likelihood of displacement of a class p target between two consecutive frames. The transition probabilities in (15) define a finite state machine (FSM) that specifies the dynamics of the centroid for the class p target. Example: Targets with constant mean velocity A particular dynamic model of interest is a target where the nominal velocity is constant. We perturb this mean or nominal velocity by an mth order random walk fluctuation. The target centroid position at instant n is then given by zn+1 = zn + d + εn
(16)
where d is the mean velocity, and εn is a discrete-valued, zero-mean white noise component that is independent of the centroid position and takes values on the discrete set S = {−m, . . . − 1, 0, 1, . . . , m} for some m ≥ 1. Figure 1 shows the central section of the FSM that corresponds to the model in (16) when d = 0 and m = 1. This FSM is simply a first order discrete Markov chain. As mentioned
1-r-q q
q i
i-1 r
i+1 r
Fig. 1. Example of Finite State Machine Diagram
before, a target that is present moves to the absent state whenever its centroid is outside the lattice Lp = {l: − lsp + 1 ≤ l ≤ L + lip }. When no target of a given class is present, we assume that there is DRAFT
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a non-zero probability pa of a new target from that same class appearing randomly at the next sensor scan. We assume that the target centroid may appear at any pixel of the centroid lattice Lp , with equal probability pa / (L + lip + lsp ). This assumption is a worst case scenario, when the detector/tracker has no a priori information about initial position of a new target. Other more elaborate distributions for the probability of reappearance are easily taken into account. III. Optimal Bayes Multitarget Detector/Tracker We assume that at each scan n an unknown number of targets ranging from zero to M may be present. The targets that are present belong to distinct classes (i.e., in the context of this model, they have different signatures). We collect the observation scans from instant 0 up to instant n in the long observation vector T Y0n = y0T . . . ynT . Given Y0n , we want to perform three tasks at instant n : (1) determine how many targets are present/absent (detection); (2) assign the detected targets to a given class (data association); (3) estimate the positions of the detected targets (tracking). A. Nonlinear Stochastic Filtering Approach As mentioned in section II, the vector
T Zn = zn1 . . . znM ,
collects the positions of the centroids of the M possible targets in the sensor image . If all
(17) znp
=
L + lip + 1,
1 ≤ p ≤ M , then no target is present in the surveillance space at the nth scan. The optimal Bayes solution to the joint detection/tracking problem is obtained by computing at each scan the joint posterior probability, P (Zn | Y0n ), i.e., the conditional probability of the vector The Bayes detector/tracker that we present processes the observations as they become available. It computes recursively P (Zn | Y0n ) at each scan, thus avoiding having to store all the measurements from instant zero up to the present. The recursion is divided into two steps. The first step is the prediction step: it uses the statistical description of the target motion between two consecutive scanned frames to predict the current position of the targets based on all past observations. Once a new sensor frame is available, a second step, known as the filtering step, uses the new measurements to correct the prediction. The incoming sensor data is processed using the information in the clutter and target signature models. In the sequel, we describe both steps in further detail. The following assumptions are made in the derivation of the algorithm: 1. In each frame, only one target from each of the M possible classes may be present. 2. The sequence of clutter frames {vn }, n ≥ 1, is independent, identically distributed (i.i.d.). 3. The sequence of target states {Zk }, k ≥ 0, is statistically independent of the sequence of clutter frames {vk }, k ≥ 0. 4. Targets from different classes move independently and the translational motions for targets from each class are described by first-order discrete Markov Chains completely specified by the transition probabilities P (z p | z p ), 1 ≤ p ≤ M , z p ∈ L˜p . n
n−1
n
5. In all observed frames, the target signatures are deterministic and known (but not necessarily timeinvariant) for each target class.
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We make the following remarks regarding the previous assumptions: a) The detector/tracker algorithm can be easily modified to account for unknown, random target signatures in each sensor frame. We discuss the necessary modifications in subsection III-D. b) Instead of assuming that at most one target from each class is present in each frame, we could have used an alternative problem setup in which there is a known maximum number of targets, Np ≥ 1, for each target class p. In this paper, for convenience, and without loss of generality, we make Np = 1, ∀p, 1 ≤ p ≤ M. c) The assumption that the sequence {vn }, n ≥ 0, is i.i.d. is equivalently to ignoring all interframe statistical correlation between the clutter pixels. The 2D GMrf model in section II-B assumes however an intraframe or spatial clutter correlation. We now detail the derivation of the algorithm. In the subsequent derivation, we denote the probability mass function of discrete-valued random variables by the capital letter P , whereas the probability density function of continuous-valued random variables is denoted by lowercase p. Prediction Step This step computes the prediction posterior probability P (Zn | Yn−1) = P (z 1 , . . . , . . . z M | Yn−1) z p ∈ L˜p 0
From P (Zn |
Y0n−1),
n
n
0
n
1≤p≤M .
(18)
we can obtain the marginal posterior probabilities of the centroid position of each
target conditioned on the past frames from instant 0 to instant n − 1. We also obtain the posterior probabilities of absence of each target conditioned on the past observations. Combining the theorem of Total Probability with Bayes law, we write X P (Zn | Y0n−1 ) = P (Zn, Zn−1 | Y0n−1 ) Zn−1
=
X
P (Zn | Zn−1, Y0n−1)P (Zn−1 | Y0n−1 ) .
(19)
Zn−1
Since the sequence of target centroid positions {Zk }, k ≥ 1, is, by assumption, a first-order Markov process, then, conditioned on Zn−1, the current state Zn is statistically independent of the sequence {Zk }, 0 ≤ k ≤ n − 2. If we add the assumption that Zn is also independent of the sequence of previous clutter frames {vk }, 0 ≤ k ≤ n − 1, n ≥ 1, we conclude that, conditioned on Zn−1, Zn is statistically independent of the previous observations, Y0n−1, i.e., P (Zn | Zn−1, Y0n−1 ) = P (Zn | Zn−1) . Replacing (20) in (19), we get P (Zn | Y0n−1) =
X
(20)
P (Zn | Zn−1) P (Zn−1 | Y0n−1) .
(21)
Zn−1
Finally, assuming that the different targets move according to statistically independent Markov chains, 1 M P (Zn | Zn−1) = P (zn1 | zn−1 ) . . . P (znM | zn−1 )
and we write P (Zn | Y0n−1) =
X
...
1 zn−1
X
(22)
1 M P (zn1 | zn−1 ) . . . P (znM | zn−1 ) P (Zn−1 | Y0n−1) .
(23)
M zn−1
Filtering Step We now compute the filtering posterior probability, P (Zn | Y0n ). From Bayes’ law, P (Zn | Y0n )
=
P (Zn | yn , Y0n−1)
(24)
=
Cnp(yn | Zn , Y0n−1)P (Zn | Y0n−1)
(25)
=
Cnp(yn | Zn )P (Zn | Y0n−1)
(26) DRAFT
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where Cn is a normalization constant. To write equation (26), we used the fact that the sequence of ∞
∞
clutter vectors {vn }n=0 is i.i.d and independent of the sequence of state vectors {Zn }n=0. Hence, given Zn , vector yn is independent of Y0n−1. The term p(yn | Zn ) is referred to in the nonlinear stochastic filtering literature as the observations kernel [29], [31] and specifies the conditional statistics of the observed data assuming that the targets’ states (positions) are known. The analytical expression for the observation kernel depends on the clutter and target models. We present next the optimal detection and tracking algorithms. B. Minimum probability of error Bayes detector For each of the M possible targets, there are two possible detection states during the nth scan: absent or present. The detection algorithm is therefore a statistical test that, based on all present and past observed data, Y0n , chooses one among 2M possible hypotheses Hm , 0 ≤ m, ≤ 2M − 1. In this notation, hypothesis H0 stands for “all M possible targets are absent”. Conversely, hypothesis H2M −1 means that all M possible targets are present. Hypotheses Hm , m 6= 0 and m 6= 2M − 1, represent all other combinations in between of presence and absence of the multiple targets. Given P (Zn | Y0n ), we compute the posterior probabilities of the detection hypothesis Hm , 0 ≤ m ≤ 2M − 1.h The minimum probability of error detector decides that hypothesis Hm is true if [32] P (Hm | Y0n ) > P (Hk | Y0n ) ∀k 6= m, 0 ≤ m, k ≤ 2M − 1 (27) n where P (Hm | Y0 ) is the posterior probability of hypothesis Hm . We describe two illustrative examples. Example 1: Single Target With a single target, there are only two possible hypotheses at each sensor scan: 1. H0: {target absent}. 2. H1: {target present}. The minimum probability of error detector assuming equal cost assignment to misses and false alarms and zero cost assignment to correct decisions reduces to H P (H0 | Y0n ) >0 1. P (H1 | Y0n ) < H1 Introducing the posterior probability vector, fn|n, such that its kth component is fn|n (k) = P (zn1 = k | Y0n ) k ∈ Le1 then P (H0 | Y0n ) = fn|n(L + li1 + 1)
(28) (29) (30)
P (H1 | Y0n ) = 1 − fn|n(L + li1 + 1) . (31) Remark: The decision rule in (28) minimizes the total probability of decision errors, misses, and false alarms. Alternatively, if we change the threshold in (28) and vary it over a wide range, the detection algorithm operates as a Neyman-Pearson detector [32] that maximizes the probability of detection for a given probability of false alarm. Example 2: Two Targets We illustrate next how to compute the quantities P (Hm | Y0n ) from the filtering posterior probability P (Zn | Y0n ) when there are two targets, i.e., M = 2. With two targets, there are 4 possible hypotheses for the presence or absence of targets at the nth sensor scan: DRAFT
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H0 = {Both targets absent}. H1 = {Target 1 absent and Target 2 present}. H2 = {Target 1 present and Target 2 absent}. H3 = {Both targets present}. We introduce the filtering posterior probability matrix, Fn|n, whose (k, j) element is the conditional probability that target 1 is at pixel k and target 2 is at pixel j, conditioned on the observation path Y0n , i.e., Fn|n(k, j) = P (zn1 = k, zn2 = j | Y0n )
e1, j ∈ L e2 . k∈L
(32)
The posterior probabilities of the different hypothesis are computed as follows: P (H0 | Y0n ) = Fn|n(L + li1 + 1, L + li2 + 1) L+l2i
P (H1 |
Y0n )
=
X
Fn|n(L + li1 + 1, j)
j=−l2s +1 L+l1i
P (H2 | Y0n ) =
X
Fn|n(k, L + li2 + 1)
k=−l1s +1 L+l1i
P (H3 | Y0n ) =
X
L+l2i
X
Fn|n(k, j) .
(33)
k=−l1s +1 j=−l2s +1
The posterior probability of the two targets being present can be alternatively calculated as 2 X P (H3 | Y0n ) = 1 − P (Hr | Y0n ) .
(34)
r=0
C. Tracking: maximum a posteriori (MAP) tracker We examine next the solution to the tracking (localization) problem. We use a maximum a posteriori (MAP) strategy that gives optimal localization in a Bayesian sense, with respect to a cost function that assigns uniform penalty to any tracking error regardless of the magnitude of the error [32]. If, after detection, hypothesis Hm , 1 ≤ m ≤ 2M − 1, is declared true, we introduce the conditional probability tensor Πm n|n defined as
P (Zn, Hm | Y0n ) n Πm . (35) n|n (Zn ) = P (Zn | Hm , Y0 ) = P (Hm | Y0n ) m The MAP Bayes tracker looks for the maximum of Πn|n over Zn to estimate the positions of the targets
that are assumed present under hypothesis Hm . Example 1: Single Target In the single target case, the tensor Π1n|n reduces to a vector whose general element is Π1n|n(k) = P (zn = k | target is present, Y0n ) fn|n(k) . (36) 1 − fn|n(L + li1 + 1) When the target is present, the maximum a posteriori (MAP) estimates of the actual target position are 1 zd = arg max Π1n|n(k) . (37) n|n =
Example 2: Two Targets
−l1s +1≤k≤L+l1i
In the case of two targets, the conditional probability tensors Πkn|n, k = 1, 2, 3 are matrices. Let Fn|n be the filtering posterior probability matrix defined in (32) and let H1, H2 and H3 be the three possible “target present” hypotheses as described before. We have three cases: DRAFT
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1. Target 1 is declared absent and target 2 is declared present: In this case, we find the optimal MAP 2 estimate of the centroid position of the class 2 target, denoted by zd n|n , using the expression Fn|n(L + li1 + 1,j) 2 zd . (38) n|n = arg max P (H1 | Y0n ) j∈L2 2. Target 1 is declared present and target 2 is declared absent: This situation is the dual of the previous 1 , is given case. The optimal MAP estimate of the centroid position of the class 1 target, denoted by zd n|n
by Fn|n(k,L + li2 + 1) 1 zd = arg max . (39) n|n P (H2 | Y0n ) k∈L1 3. Targets 1 and 2 are declared present: when both targets are declared present, the optimal MAP centroid estimates are d 1 2 (zd n|n , zn|n ) = arg
Fn|n(k, j) max n k∈L1 , j∈L2 P (H3 | Y0 )
k ∈ L1, j ∈ L2 .
(40)
D. Detection/Tracking of targets with random signature In the previous subsections, we considered the situation where the targets’ signatures are deterministic and known. We now extend the algorithm to account for targets with random pixel intensity. For simplicity, assume that the targets have equal size, i.e., lip = li and lsp = ls , 1 ≤ p ≤ M , where M is the number of targets. Let l = li + ls + 1 and define the l-dimensional column vector of the signature parameters apn such that its ith component is apn (i) = api,n
− li ≤ i ≤ ls
1≤p≤M .
Now stack these signature parameters in the column vector T T Θn = (a1n )T . . . (aM n )
(41) (42)
Assume that the sequence {Θn } is i.i.d. and independent of {Zn } and {vn}, for n ≥ 1. After a few algebraic steps, it is easy to show thatZ n P (Zn | Y0 ) = Cn p(yn | Θn, Zn)p(Θn ) dΘn P (Zn | Y0n−1 ) .
(43)
Equation (43) shows that, under the assumptions that the sequence of target signatures {Θn } is i.i.d. and statistically independent of both the sequence of target positions and the sequence of clutter frames, we can obtain the observations kernel for each possible state vector Zn at instant n by averaging the conditional pdf of the measurements p(yn |Θn, Zn ) over all possible realizations of the vector of target signatures. E. Comparison with dynamic programming approaches In this subsection, we contrast the nonlinear stochastic filtering approach to target tracking with previous work by Barniv [3]. We contrast Barniv’s paper [3] with ours with respect to two issues: (i) the Viterbi algorithm used in [3] versus Bayes’ law as used by us; (ii) setup of the problem and other modeling assumptions. We also make some brief comments on computational complexity. (i) Viterbi algorithm versus Bayes’ law For simplicity of notation and to follow Barniv’s model, we consider a single target scenario and assume initially that the target is present in the surveillance space in all observed sensor frames, so that no hard detection decision between presence or absence of target has to be made at each sensor scan. Reference [3] applies Bellman’s dynamic programming [12] and its DRAFT
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implementation by Viterbi [10] and Larson [11] to solve the target trajectory estimation problem. Let zk and yk denote respectively the unknown target state and the observations at instant k. Define the target state path, Zk0 , and the observation path, Y0k , such that Zk0 = [z0 , z1 , . . . , zk ] Denote by zbk+1|k+1
(44)
Y0k = [y0 , y1 , . . . yk ] . (45) the estimate of the unknown target state zk+1 at instant k + 1, based on the observa-
tions Y0k+1 . Barring some minor differences in the indexing of the state and observation paths, Barniv’s
estimate of the target state is given by zbk+1|k+1 = arg max I(zk+1 ) zk+1 where I(zk ) is a quantity that is proportional to max P (Zk0 | Y0k )
(46) (47)
Zk−1 0
and is computed using the recursion I(zk+1 ) = p(yk+1 | zk+1 ) max [P (zk+1 | zk )I(zk )] k≥0. (48) zk When the observation y0 at instant zero is available, equation (48) is initialized with I(z0 ) = p(y0 | z0) P (z0 ) . (49) Barring some minor differences in the initialization of the algorithm due to the availability of the observation y0 , equation (48) corresponds essentially to the forward recursion step of the Viterbi algorithm, see [10]. By contrast, our tracking algorithm is an MAP estimator based on Bayes’ law, i.e., our estimate for the unknown target state at instant k + 1 is zbk+1|k+1 = arg max P (zk+1 | Y0k+1 ) . (50) zk+1 Combining the prediction and filtering steps of our algorithm in one equation, the posterior probability mass function on the righthand side of equation (50)" is obtained by the recursion# X P (zk+1 | Y0k+1 ) = Ck+1 p(yk+1 | zk+1 ) P (zk+1 | zk )P (zk | Y0k )
k≥0
(51)
where Ck+1 is a normalization constant that is independent of zk+1 . We initialize (51) with P (z0 | y0 ) = C0p(y0 | z0 )P (z0 ) where C0 is a normalization constant that is independent of z0 .
(52)
zk
Equations (48) and (51) clearly define two different recursive algorithms. We now show that equations (46) and (50) correspond to two different maximization problems and may lead to different state estimates. Write P (Zk+1 | Y0k+1) = P (Zk0 , zk+1 | Y0k+1) 0 = P (Zk0 | zk+1, Y0k+1) P (zk+1 | Y0k+1 ) . (53) The second factor in equation (53) is the conditional pmf of the current state zk+1 given the path of observations Y0k+1 up to instant k + 1. This is what our proposed nonlinear stochastic filter computes at each instant. The first factor can be simplified to P (Zk0 | zk+1 , Y0k+1) = P (Zk0 | zk+1 , Y0k ) . Recall that Barniv’s state estimate is given by ( )
(54)
zbk+1|k+1 = arg max max P (Z0k+1 | Y0k+1 ) zk+1
(
Zk 0
= arg max max P (Zk0 | zk+1 , Y0k ) P (zk+1 | Y0k+1 ) zk+1
Zk 0
)
.
(55)
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Note now(that max zk+1
max P (Zk0 Zk 0
|
zk+1 , Y0k ) P (zk+1
Y0k+1 )
|
)
6=
"
max P (Zk0 Z0k
|
Y0k )
#
max P (zk+1 | Y0k+1) . zk+1
16
(56)
If the factorization in (56) were possible, then Barniv’s estimate zbk+1| k+1 and ours would coincide. How-
ever, because the maximization on the lefthand side of (56) does not factor as the expression on the righthand side of the same equation, the two estimates may be different. Also note that we provide in our paper only the filtering estimate for the unknown state path, i.e., our algorithm computes the sequence zbk|k
for k ≥ 0 .
Reference [3] on the other hand provides the smoothed state path estimate, i.e., the sequence zbi| k
for k ≥ 0, i ≤ k .
The smoothed estimates in [3] are obtained using the backward retrieval step of the Viterbi algorithm, see [10]. In terms of applications, Barniv’s algorithm provides a batch estimate of the state path, ck = arg max P (Zk | Yk ), Z 0
Zk 0
0
0
whereas ours is an on-line algorithm that is similar in nature to Kalman-Bucy filtering, i.e., whenever a new state estimate is available at instant k, we do not go back and reestimate the previous states zi for i ≤ k. Finally, in a multitarget scenario where targets are not assumed a priori to be always present, a multitarget detection step must be added to the tracking algorithm. In Barniv’s work, the Viterbi forward recursion is run as if only one single target were present and multitarget detection is done simply by thresholding the function I(zk ) at the last stage of the recursion. All states zk for which I(zk ) exceeds a certain threshold are assumed to be the final state of one possible target. The state trajectories for each detected target are then retrieved by moving backwards along the path of corresponding surviving nodes in the Viterbi trellis, see [10] and [3] for details. Since this procedure leads to a large number of false detections (roughly 40 detections per target [3]), a post-processing clustering step is used to merge nearby estimated trajectories. In our approach, we expand the state space to include dummy “absent target” states and propagate the joint posterior probability mass function of all target states, including the dummy states. Multitarget detection is then accomplished using a minimum probability of error M-ary Bayes hypotheses test. (ii) Setup of problem and modeling assumptions In the sequel, we contrast briefly the state and observation models used in our work with the models introduced in [3]. In our paper, for targets that are present, the corresponding states are the pixel locations at each sensor frame of the target centroids in the discrete centroid grid. In [3], the states are defined as straight line trajectory segments across a group of G > 1 sensor frames that define the stages (instants) for the Viterbi forward recursion. The corresponding observation (measurements) model in [3] involves a differential pre-processing of the original sensor images. After pre-processing, it is assumed in [3] that all residual measurement noise is Gaussian and white. In our work, the measurements are the raw sensor frames themselves, with no pre-processing except for a possible removal of the moving local mean (as explained in section V-B). Instead of using a white Gaussian measurement noise assumption, we take full advantage of the real statistics of the background clutter
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to improve detection/tracking performance. That includes exploring both the clutter spatial (intraframe) correlation and the clutter’s possibly non-Gaussian amplitude statistics. (iii) Computational complexity We compare the computational complexity of the Viterbi algorithm to e our proposed Bayesian tracker. Let zn ∈ Le be the hidden variable, with the number of elements in L being denoted by L1 . Define the L1 -dimensional vectors fk|k and ik , such that fk|k (j) = P (zn = j | Y0k ), e where I(zk ) is the function defined in (47). Introduce also the L1 × L1 and ik (j) = I(zk = j), j ∈ L,
e The transition probability matrix, PT , such that PT (n, j) = P (zk+1 = n, | zk = j), (k, j) ∈ Le × L.
recursion in equation (51), that corresponds to the Bayes tracker, can be rewritten in matrix notation as fk+1| k+1 = Ck+1 Sk+1 PT fk|k (57) where denotes the pointwise multiplication operator and Sk+1 is a L1 × 1 vector such that Sk+1 (j) = e On the other hand, Viterbi’s forward recursion in equation (48) is written as p(yk+1 | zk+1 = j), j ∈ L. ik+1 = Sk+1 max PlT ik 1≤l≤L (58) 1 l l e The where P is the lth row of the transition matrix PT , i.e., P (j) = P (zk+1 = l | zk = j), j ∈ L. T
T
bracketed expression on the righthand side of equation (58) reads as follows: for each l, 1 ≤ l ≤ L1 , do the pointwise multiplication of the lth row of the transition probability matrix, PT , by the previous filtering vector, ik , resulting in a L1-dimensional vector, ilk . Then, look for the maximum of the entries ilk (j) over the range 1 ≤ j ≤ L1 and assign this maximum to the lth entry of the bracketed vector. A comparison between equations (57) and (58) shows that the two recursions differ basically in the computation of the bracketed vector on the righthand side. The Bayesian tracker involves the multiplication of an L1 × L1 matrix by a L1 × 1 vector, which requires L21 floating point multiplications and L1 (L1 − 1) floating point additions. On the other hand, the forward recursion of the Viterbi algorithm requires L21 floating point multiplications and L1 global maximum searches over an L1 -dimensional vector. Those maximum searches require in turn L1 (L1 − 1) comparisons. The two algorithms therefore trade arithmetic (addition) computational complexity for logic (comparison) computational complexity. We make two additional remarks: Remark 1: The Viterbi smoother requires that, in addition to the forward propagation of I(zk ) using recursion (58), we must also store the indices of the maxima over j, 1 ≤ j ≤ L1 , of ikl (j), for all k > 0 and all l, 1 ≤ l ≤ L1 . This table of stored indices is necessary for the implementation of Viterbi’s backward retrieval step, see [10]. Remark 2: In most applications, the transition probability matrix, PT , is not a full L1 × L1 matrix, as transitions are only alllowed between adjacent target states. As a result of the sparse nature of PT , the number of floating point multiplications required in the prediction step for both the Bayes tracker and the Viterbi recursion falls in practice from O(L21 ) to O(αL1), where α L. end of loop • for I, J = 1 to L1 i = I − ls ; j = J − ls ; Matched filter: λi, j = ai,j =
P P k
l
ak, l µi+k, j+l , with the limits for the summations given in Table III.
ak, l , with k and l in the ranges assigned to each pair (i, j) in Table III.
At = Ir ⊗ (Ir − βh Hl ) − βv Hr ⊗ Il where (r, l) = size(ai,j ). Energy term: ρi,j = (vec [ai, j ])T At (vec [ai, j ]).
2 . Observations kernel: Sn ((I − 1)L1 + J) = exp (2λI−ls, J−ls − ρI−ls J−ls )/2 σu
end of loop. • Normalized kernel entry for absent target state: Sn (L21 + 1) = 1.
P
• Filtering step: fn|n = Cn Sn fn|n−1 where Cn is a normalization constant such that f (l) = 1. l n|n H0 • Binary Detection: fn|n (L21 + 1) > 1 − fn|n (L21 + 1). < H1 • MAP estimation: If hypothesis H1 (target present) declared true, zˆn|n = arg maxl∈L fn|n (l). • fn−1|n−1 = fn|n . End of outer for-loop c) End of program. TABLE II Pseudocode for the 2D Bayes detector/tracker
λ(i, j) −ls + 1 ≤ i ≤ li l i + 1 ≤ i ≤ L − ls L − ls + 1 ≤ i ≤ L + li
−ls + 1 ≤ j ≤ li Pls Pls k=−i+1
Pls
k=−li
PL−i
k=−li
l=−j+1 (.)
Pls
l=−j+1 (.)
Pls
l=−j+1 (.)
li + 1 ≤ j ≤ L − ls Pls Pls k=−i+1 l=−li (.) Pls Pls k=−li l=−li (.) PL−i Pls k=−li l=−li (.)
L − ls + 1 ≤ j ≤ L + li Pls PL−j l=−li (.) k=−i+1 Pls PL−j k=−li l=−li (.) PL−i PL−j k=−li l=−li (.)
TABLE III Computation of the data term λij
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Remark The actual implementation of the matrix multiplication PT fn−1|n−1 in Table II explores the sparse and block-banded structure of the transition probability matrix PT . Note also that the energy term ρ is constant for the range li + 1 ≤ i, j ≤ L − ls and, therefore, can be computed off-line. In general, for an L × L sensor grid, it can be shown that, using the GMrf clutter model, the Markov chain motion model, and the small extended target models, we reduce total number of required floating point multiplications from O(L6 ) to O(αL2 ) in the filtering step of the algorithm and, from O(L4 ) to O(γL2 ) in the prediction step, where γ > li + ls , this cost is dominated by the computation of λi,j that is an operation of order O(βL2 ) with β
